Introduction to Finite Elements in Engineering

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This book provides an integrated approach to finite element methodologies. The development of finite element theory is combined with examples and exercises involving engineering applications. The steps used in the development of the theory are implemented in complete, self-contained computer programs. While the strategy and philosophy of the previous editions has been retained, the Third Edition has been updated and improved to include new material on additional topics.Chapter topics cover fundamental concepts, matrix algebra and gaussian elimination, one-dimensional problems, trusses, two-dimensional problems using constant strain triangles, axisymmetric solids subjected to axisymmetric loading, two-dimensional isoparametric elements and numerical integration, beams and frames, three-dimensional problems in stress analysis, scalar field problems, dynamic considerations, and preprocessing and postprocessing.For practicing engineers as a valuable learning resource.

Preface

xv

Fundamental Concepts

1

(21)

Introduction

1

(1)

Historical Background

1

(1)

Outline of Presentation

2

(1)

Stresses and Equilibrium

2

(2)

Boundary Conditions

4

(1)

Strain-Displacement Relations

4

(2)

Stress-Strain Relations

6

(2)

Special Cases

7

(1)

Temperature Effects

8

(1)

Potential Energy and Equilibrium; The Rayleigh-Ritz Method

9

(4)

Potential Energy II

9

(2)

Rayleigh-Ritz Method

11

(2)

Galerkin's Method

13

(3)

Saint Venant's Principle

16

(1)

Von Mises Stress

17

(1)

Computer Programs

17

(1)

Conclusion

18

(4)

Historical References

18

(1)

Problems

18

(4)

Matrix Algebra and Gaussian Elimination

22

(23)

Matrix Algebra

22

(7)

Row and Column Vectors

23

(1)

Addition and Subtraction

23

(1)

Multiplication by a Scalar

23

(1)

Matrix Multiplication

23

(1)

Transposition

24

(1)

Differentiation and Integration

24

(1)

Square Matrix

25

(1)

Diagonal Matrix

25

(1)

Identity Matrix

25

(1)

Symmetric Matrix

25

(1)

Upper Triangular Matrix

26

(1)

Determinant of a Matrix

26

(1)

Matrix Inversion

26

(1)

Eigenvalues and Eigenvectors

27

(1)

Positive Definite Matrix

28

(1)

Cholesky Decomposition

29

(1)

Gaussian Elimination

29

(10)

General Algorithm for Gaussian Elimination

30

(3)

Symmetric Matrix

33

(1)

Symmetric Banded Matrices

33

(2)

Solution with Multiple Right Sides

35

(1)

Gaussian Elimination with Column Reduction

36

(2)

Skyline Solution

38

(1)

Frontal Solution

39

(1)

Conjugate Gradient Method for Equation Solving

39

(6)

Conjugate Gradient Algorithm

40

(1)

Problems

41

(2)

Program Listings

43

(2)

One-Dimensional Problems

45

(58)

Introduction

45

(1)

Finite Element Modeling

46

(2)

Element Division

46

(1)

Numbering Scheme

47

(1)

Coordinates and Shape Functions

48

(4)

The Potential-Energy Approach

52

(4)

Element Stiffness Matrix

53

(1)

Force Terms

54

(2)

The Galerkin Approach

56

(2)

Element Stiffness

56

(1)

Force Terms

57

(1)

Assembly of the Global Stiffness Matrix and Load Vector

58

(3)

Properties of K

61

(1)

The Finite Element Equations; Treatment of Boundary Conditions

62

(16)

Types of Boundary Conditions

62

(1)

Elimination Approach

63

(6)

Penalty Approach

69

(5)

Multipoint Constraints

74

(4)

Quadratic Shape Functions

78

(6)

Temperature Effects

84

(19)

Input Data File

88

(1)

Problems

88

(10)

Program Listing

98

(5)

Trusses

103

(27)

Introduction

103

(1)

Plane Trusses

104

(10)

Local and Global Coordinate Systems

104

(1)

Formulas for Calculating l and m

105

(1)

Element Stiffness Matrix

106

(1)

Stress Calculations

107

(4)

Temperature Effects

111

(3)

Three-Dimensional Trusses

114

(2)

Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions

Interfacing with Previous Finite Element Programs and a Program for Determining Critical Speeds of Shafts

391

(1)

Guyan Reduction

392

(2)

Rigid Body Modes

394

(2)

Conclusion

396

(15)

Input Data File

397

(2)

Problems

399

(5)

Program Listings

404

(7)

Preprocessing and Postprocessing

411

(29)

Introduction

411

(1)

Mesh Generation

411

(8)

Region and Block Representation

411

(1)

Block Corner Nodes, Sides, and Subdivisions

412

(7)

Postprocessing

419

(5)

Deformed Configuration and Mode Shape

419

(1)

Contour Plotting

420

(1)

Nodal Values from Known Constant Element Values for a Triangle

421

(2)

Least Squares Fit for a Four-Noded Quadrilateral

423

(1)

Conclusion

424

(16)

Input Data File

425

(1)

Problems

425

(2)

Program Listings

427

(13)

Appendix Proof of dA = det J dξ dη

440

(3)

Bibliography

443

(4)

Answers to Selected Problems

447

(2)

Index

449

The first edition of this book appeared over 10 years ago and the second edition followed a few years later. We received positive feedback from professors who taught from the book and from students and practicing engineers who used the book. We also benefited from the feedback received from the students in our courses for the past 20 years. We have incorporated several suggestions in this edition. The underlying philosophy of the book is to provide a clear presentation of theory, modeling, and implementation into computer programs. The pedagogy of earlier editions has been retained in this edition.New material has been introduced in several chapters. Worked examples and exercise problems have been added to supplement the learning process. Exercise problems stress both fundamental understanding and practical considerations. Theory and computer programs have been added to cover acoustics, axisymmetric quadrilateral elements, conjugate gradient approach, and eigenvalue evaluation. Three additional programs have now been introduced in this edition. All the programs have been developed to work in the Windows environment. The programs have a common structure that should enable the users to follow the development easily. The programs have been provided in Visual Basic, Microsoft Excel/Visual Basic, MATLAB, together with those provided earlier in QBASIC, FORTRAN and C. The Solutions Manual has also been updated.Chapter 1 gives a brief historical background and develops the fundamental concepts. Equations of equilibrium, stress-strain relations, strain-displacement relations, and the principles of potential energy are reviewed. The concept of Galerkin's method is introduced.Properties of matrices and determinants are reviewed in Chapter 2. The Gaussian elimination method is presented, and its relationship to the solution of symmetric banded matrix equations and the skyline solution is discussed. Cholesky decomposition and conjugate gradient method are discussed.Chapter 3 develops the key concepts of finite element formulation by considering one-dimensional problems. The steps include development of shape functions, derivation of element stiffness, formation of global stiffness, treatment of boundary conditions, solution of equations, and stress calculations. Both the potential energy approach and Galerkin's formulations are presented. Consideration of temperature effects is included.Finite element formulation for plane and three-dimensional trusses is developed in Chapter 4. The assembly of global stiffness in banded and skyline forms is explained. Computer programs for both banded and skyline solutions are given.Chapter 5 introduces the finite element formulation for two-dimensional plane stress and plane strain problems using constant strain triangle (CST) elements. Problem modeling and treatment of boundary conditions are presented in detail. Formulation for orthotropic materials is provided. Chapter 6 treats the modeling aspects of axisymmetric solids subjected to axisymmetric loading. Formulation using triangular elements is presented. Several real-world problems are included in this chapter.Chapter 7 introduces the concepts of isoparametric quadrilateral and higher order elements and numerical integration using Gaussian quadrature. Formulation for axisymmetric quadrilateral element and implementation of conjugate gradient method for quadrilateral element are given.Beams and application of Hermite shape functions are presented in Chapter 8. The chapter covers two-dimensional and three-dimensional frames.Chapter 9 presents three-dimensional stress analysis. Tetrahedral and hexahedral elements are presented. The frontal method and its implementation aspects are discussed.Scalar field problems are treated in detail in Chapter 10. While Galerkin as well as energy approaches have been used in every chapter, with equal importance, only Galerkin's approach is used i